WOLFRAM|DEMONSTRATIONS PROJECT

Markov Volatility Random Walks

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base volatility
0.2
base trend
0.1
spike
chance
0.01
length
20
strength
3.
steps
250
total change
1.7239
1.4509
1.2728
1.0988
0.9287
annual rate of return
0.7239
0.4509
0.2728
0.0988
-0.0713
annual standard deviation
0.3947
0.3575
0.3758
0.3226
0.3176
mean daily return
0.0025
0.0017
0.0012
0.0006
-0.0001
daily standard deviation
0.0249
0.0226
0.0237
0.0204
0.0200
daily return skewness
0.3436
0.1058
-0.3835
0.9757
-0.6481
daily return kurtosis
5.5047
6.0978
8.3560
8.8767
6.1766
A decent first approximation of real market price activity is a lognormal random walk. But with a fixed volatility parameter, such models miss several stylized facts about real financial markets. Allowing the volatility to change through time according to a simple Markov chain provides a much closer approximation to real markets. Here the Markov chain has just two possible states: normal or elevated volatility. Either state tends to persist, with a small chance of transitioning to the opposite state at each time-step.